CONVEX-CONCAVE ARC GEAR MECHANISM USED FOR PARALLEL AXES TRANSMISSION

Abstract

The present invention relates to a concave-convex arc line gear mechanism for parallel shaft transmission, which comprises a driving line gear and a driven line gear, axes of the driving line gear and the driven line gear being parallel to each other to form a transmission pair. The driving line gear is consisted of convex teeth and a driving wheel body, a surface of the convex tooth comprising a pair of convex arc-shaped tooth flanks and a tooth top surface The driven line gear is consisted of concave teeth and a driven wheel body, a surface of the concave tooth comprising a pair of concave arc-shaped tooth flanks and a tooth bottom surface A meshing track of the transmission pair during transmission is a space curve. One arc-shaped tooth flank of the driving line gear and one arc-shaped tooth flank of the driven line gear present a point contact of convex arc and concave arc at a meshing point. Tooth shapes of the driving line gear and the driven line gear are interchangeable, i.e., the driving line gear has concave teeth, while the driven line gear has convex teeth. The line gear mechanism of the present invention has high transmission ratio, high contact strength, high load-bearing capacity, wide range of application, and is easy to be machined, which is especially suitable for space-limited microminiature mechanical, micro mechanical and conventional mechanical applications.

Claims

1. A concave-convex arc line gear mechanism for parallel shaft transmission, characterized in that: it comprises a driving line gear and a driven line gear, axes of the driving line gear and the driven line gear being parallel to each other to form a transmission pair; the driving line gear is consisted of convex line teeth and a driving wheel body, a surface of the convex line tooth comprising a pair of convex arc-shaped tooth flanks and a tooth top surface and the driven line gear is consisted of concave teeth and a driven wheel body, a surface of the concave tooth comprising a pair of concave arc-shaped tooth flanks and a tooth bottom surface.

2. The concave-convex arc line gear mechanism for parallel shaft transmission according to claim 1, wherein the convex tooth of the driving line gear is formed by driving tooth profile composed of two sections of arcs and a section of straight line moving along a driving contact curve and two driving tooth thickness auxiliary curves; the concave tooth of the driven line gear is formed by driven tooth profile composed of two sections of arcs and a section of straight line moving along a driven contact curve and two driven tooth thickness auxiliary curves; the driving contact curve and the driven contact curve are a pair of conjugate space curves which conform to space curve meshing equations; and a tooth profile of the convex teeth of the driving line gear and a tooth profile of the concave teeth of the driven line gear are located on normal planes of the driving contact curve and the driven contact curve, respectively.

3. The concave-convex arc line gear mechanism for parallel shaft transmission according to claim 1, wherein a meshing track of the transmission pair during transmission is a space curve; and one arc-shaped tooth flank of the driving line gear and one arc-shaped tooth flank of the driven line gear present a point contact of convex arc and concave arc at a meshing point.

4. The concave-convex arc line gear mechanism for parallel shaft transmission according to claim 1, wherein tooth shapes of the driving line gear and the driven line gear are interchangeable, i.e., the driving line gear has concave gear teeth, while the driven line gear has convex gear teeth.

5. The concave-convex arc line gear mechanism for parallel shaft transmission according to claim 1, wherein the driving wheel body and the driven wheel body are cylindrical wheel bodies; driving line teeth project from the cylindrical wheel body; and driven line teeth are recessed into the cylindrical wheel body.

6. The concave-convex arc line gear mechanism for parallel shaft transmission according to claim 2, wherein the driving contact curve is a space helical curve, an equation of which in a coordinate system o.sub.1x.sub.1y.sub.1z.sub.1 is as follows: $\begin{array}{cc}\{\begin{array}{c}{x}_M^{\left(1\right)}=m.Math..Math.\cos .Math..Math.t\\ y_M^{\left(1\right)}=m.Math..Math.\sin .Math..Math.t\\ z_M^{\left(1\right)}=n.Math..Math.+n.Math..Math.t\end{array},& \end{array}$ wherein, t is a parameter, $t\left[t_s,t_e\right],.Math..Math.t=t_e-t_s,t_s=--\frac{.Math..Math.t}{2},t_e=-+\frac{.Math..Math.t}{2},$ satisfying contact ratio condition: $=\frac{.Math..Math.t\mathrm{number}.Math..Math.\mathrm{of}.Math..Math.\mathrm{teeth}.Math..Math.\mathrm{of}.Math..Math.\mathrm{driving}.Math..Math.\mathrm{wheel}}{2.Math.}1;m$ is a helical radius of the space helical curve, n is a parameter of a pitch of the space helical curve, and the pitch p=2n; the driven contact curve is a space curve conjugate with the driving wheel contact curve, an equation of which in a coordinate system o.sub.2x.sub.2y.sub.2z.sub.2 is as follows: $\{\begin{array}{c}{x}_M^{\left(2\right)}=-\left(m-a\right).Math.\cos .Math.\frac{.Math.t+}{i_{12}}\\ y_M^{\left(2\right)}=\left(m-a\right).Math.\sin .Math..Math.\frac{t+}{i_{12}}\\ z_M^{\left(2\right)}=n.Math..Math.+n.Math..Math.t\end{array},$ wherein, i.sub.12 is a transmission ratio between the driving line gear and the driven line gear, and a is a center distance between two gears, a=(1+i.sub.12)m.

7. The concave-convex arc line gear mechanism for parallel shaft transmission according to claim 2, wherein the two driving tooth thickness auxiliary curves comprise a first driving tooth thickness auxiliary curve and a second driving tooth thickness auxiliary curve, the first driving tooth thickness auxiliary curve being located between the driving contact curve and the second driving tooth thickness auxiliary curve, with an equation of the first driving tooth thickness auxiliary curve in a coordinate system o.sub.1x.sub.1y.sub.1z.sub.1 being as follows: $\begin{array}{cc}\{\begin{array}{c}{x}_{M_{11}}^{\left(1\right)}=m.Math..Math.\cos .Math..Math.t-\frac{c_1.Math.n.Math..Math.\sin .Math..Math.t}{\sqrt{n^2+m^2}}\\ y_{M_{11}}^{\left(1\right)}=m.Math..Math.\sin .Math..Math.t+\frac{c_1.Math.n.Math..Math.\cos .Math..Math.t}{\sqrt{n^2+m^2}}\\ x_{M_{11}}^{\left(1\right)}=n.Math..Math.+n.Math..Math.t-\frac{c_1.Math.m}{\sqrt{n^2+m^2}}\end{array},& \end{array}$ an equation of the second driving tooth thickness auxiliary curve in the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1 being as follows: $\begin{array}{cc}\{\begin{array}{c}{x}_{M_{12}}^{\left(1\right)}=m.Math..Math.\cos .Math..Math.t-\frac{2.Math.c_1.Math.n.Math..Math.\sin .Math..Math.t}{\sqrt{n^2+m^2}}\\ y_{M_{12}}^{\left(1\right)}=m.Math..Math.\sin .Math..Math.t+\frac{2.Math.c_1.Math.n.Math..Math.\cos .Math..Math.t}{\sqrt{n^2+m^2}}\\ x_{M_{12}}^{\left(1\right)}=n.Math..Math.+n.Math..Math.t-\frac{2.Math.c_1.Math.m}{\sqrt{n^2+m^2}}\end{array},& \end{array}$ wherein, 2c.sub.1 is a tooth thickness of the driving gear tooth; the two driving tooth profile arcs are symmetrical about the first driving tooth thickness auxiliary curve, and a radius of the arc is .sub.1; at a meshing point, an angle between a line which connects the meshing point and an arc center and a binormal vector of the driving contact curve is ; the driving tooth profile straight line segment and the binormal vector of the driving contact curve are parallel to each other, a distance between the driving tooth profile straight line segment and the binormal vector is h.sub.a1, h.sub.a1=h.sub.a*.sub.1(1sin ), wherein h.sub.a* is an addendum coefficient, an range of h.sub.a*being 0.80.97, and a length of the driving tooth profile straight line segment depends on the tooth thickness 2c.sub.1, the arc radius .sub.1, the angle and h.sub.a1, which is obtained by intercepting a specific distance h.sub.a1 from a straight line with two driving tooth profile arcs.

8. The concave-convex arc line gear mechanism for parallel shaft transmission according to claim 2, wherein the two driven tooth thickness auxiliary curves comprise a first driven tooth thickness auxiliary curve and a second driven tooth thickness auxiliary curve, the first driven tooth thickness auxiliary curve being located between the driven contact curve and the second driven tooth thickness auxiliary curve, an equation of the first driven tooth thickness auxiliary curve in a coordinate system o.sub.2x.sub.2y.sub.2z.sub.2 being as follows: $\{\begin{array}{c}{x}_{M_{21}}^{\left(2\right)}=-\left(m-a\right).Math.\cos .Math.\frac{.Math.t+}{i_{12}}+\frac{c_2.Math.n}{\sqrt{n^2+m^2}}.Math.\sin .Math..Math.\frac{t+}{i_{12}}\\ y_{M_{21}}^{\left(2\right)}=\left(m-a\right).Math.\sin .Math..Math.\frac{t+}{i_{12}}+\frac{c_2.Math.n}{\sqrt{n^2+m^2}}.Math.\cos .Math..Math.\frac{t+}{i_{12}}\\ z_{M_{21}}^{\left(2\right)}=n.Math..Math.+n.Math..Math.t+\frac{c_2.Math.m}{\sqrt{n^2+m^2}}\end{array},$ an equation of the second driven tooth thickness auxiliary curve in the coordinate system o.sub.2x.sub.2y.sub.2z.sub.2 being as follows: $\{\begin{array}{c}{x}_{M_{22}}^{\left(2\right)}=-\left(m-a\right).Math.\cos .Math.\frac{.Math.t+}{i_{12}}+\frac{2.Math.c_2.Math.n}{\sqrt{n^2+m^2}}.Math.\sin .Math..Math.\frac{t+}{i_{12}}\\ y_{M_{22}}^{\left(2\right)}=\left(m-a\right).Math.\sin .Math..Math.\frac{t+}{i_{12}}+\frac{2.Math.c_2.Math.n}{\sqrt{n^2+m^2}}.Math.\cos .Math..Math.\frac{t+}{i_{12}}\\ z_{M_{22}}^{\left(2\right)}=n.Math..Math.+n.Math..Math.t+\frac{2.Math.c_2.Math.m}{\sqrt{n^2+m^2}}\end{array},$ wherein, 2c.sub.2 is a tooth thickness of the driven gear tooth; the two driven tooth profile arcs are symmetrical about the middle first driven tooth thickness auxiliary curve, a radius of the arc is .sub.2; at a meshing point, an angle between a line which connects the meshing point and an arc center and a binormal vector of the driven contact curve is ; the driven tooth profile straight line segment and the binormal vector of the driven contact curve are parallel to each other, a distance between the driven tooth profile straight line segment and the binormal vector is h.sub.f2, h.sub.f2=h.sub.f*h.sub.a1, wherein h.sub.f* is a dedendum coefficient, a range of h.sub.f*being 1.42, and a length of the driven tooth profile straight line segment depends on the tooth thickness 2c.sub.2, the arc radius .sub.2, the angle and h.sub.f2, which is obtained by intercepting a specific distance h.sub.f2 from a straight line with two driven tooth profile arcs.

9. The concave-convex arc line gear mechanism for parallel shaft transmission according to claim 5, wherein a diameter of the driving wheel body is d.sub.f1, with a valued d.sub.f1=2m2(h.sub.a1+d.sub.f*), wherein d.sub.f* is a clearance coefficient, a range of d.sub.f*being 0.52; and a diameter of the driven wheel body is d.sub.a2, with a value d.sub.a2=2(am)+2h.sub.a1.

10. The concave-convex arc line gear mechanism for parallel shaft transmission according to claim 7 wherein the arc radius .sub.1 of the two driving tooth profile arcs, the arc radius .sub.2 of the two driven tooth profile arcs and the angle satisfy following conditions: .sub.2=k.sub.1; $\mathrm{if}.Math..Math.\left[30.Math.,40.Math.\right],\mathrm{then}.Math..Math.{}_1=\frac{1.1.Math.c_1}{\cos .Math..Math.},k\left(0,\frac{1}{4}\right);\mathrm{or}.Math..Math.{}_1=\frac{1.2.Math.c_1}{\cos .Math..Math.},.Math.k\left(0,\frac{1}{5}\right);\mathrm{or}.Math..Math.{}_1=\frac{1.4.Math.c_1}{\cos .Math..Math.},k\left(0,\frac{1}{7}\right);\mathrm{or}.Math..Math.{}_1=\frac{1.6.Math.c_1}{\cos .Math..Math.},k\left(0,\frac{1}{11}\right);$ $\mathrm{if}.Math..Math.\left(40.Math.,45.Math.\right],\mathrm{then}.Math..Math.{}_1=\frac{1.4.Math.c_1}{\cos .Math..Math.},k\left(0,\frac{1}{7}\right);\mathrm{or}.Math..Math.{}_1=\frac{1.6.Math.c_1}{\cos .Math..Math.},.Math.k\left(0,\frac{1}{11}\right).$

11. The concave-convex arc line gear mechanism for parallel shaft transmission according to claim 8 wherein the arc radius .sub.1 of the two driving tooth profile arcs, the arc radius .sub.2 of the two driven tooth profile arcs and the angle satisfy following conditions: ${}_2=k.Math..Math.{}_1;$ $\mathrm{if}.Math..Math.\left[30.Math.,40.Math.\right],\mathrm{then}.Math..Math.{}_1=\frac{1.1.Math.c_1}{\cos .Math..Math.},k\left(0,\frac{1}{4}\right);\mathrm{or}.Math..Math.{}_1=\frac{1.2.Math.c_1}{\cos .Math..Math.},.Math.k\left(0,\frac{1}{5}\right);\mathrm{or}.Math..Math.{}_1=\frac{1.4.Math.c_1}{\cos .Math..Math.},k\left(0,\frac{1}{7}\right);\mathrm{or}.Math..Math.{}_1=\frac{1.6.Math.c_1}{\cos .Math..Math.},k\left(0,\frac{1}{11}\right);.Math.\mathrm{if}.Math..Math.\left(40.Math.,45.Math.\right],\mathrm{then}.Math..Math.{}_1=\frac{1.4.Math.c_1}{\cos .Math..Math.},k\left(0,\frac{1}{7}\right);\mathrm{or}.Math..Math.{}_1=\frac{1.6.Math.c_1}{\cos .Math..Math.},.Math.k\left(0,\frac{1}{11}\right).$

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0028] FIG. 1 is a schematic view of a concave-convex arc line gear mechanism for parallel shaft transmission according to an embodiment of the present invention.

[0029] FIG. 2 is a schematic view of the driving contact curve and driven contact curve meshing at point M in FIG. 1.

[0030] FIG. 3 is a left view of the driving line gear in FIG. 1.

[0031] FIG. 4 is a three-dimensional schematic view of A-A section of the driving line gear in FIG. 3 sectioned by a normal plane P of the driving contact curve.

[0032] FIG. 5 is a right view of the driven wheel in FIG. 1.

[0033] FIG. 6 is a three-dimensional schematic view of B-B section of the driven line gear in FIG. 5 sectioned by a normal plane P of the driven contact curve.

[0034] FIG. 7 is a three-dimensional schematic view of a section of the driving line gear and the driven line gear in FIG. 1 sectioned by a normal plane at the meshing point.

[0035] FIG. 8 is a partial enlarged view of a section of the driving line gear and the driven line gear in FIG. 7 sectioned by a normal plane at the meshing point.

[0036] In the above figures: 1tooth flank of driving line gear, 2tooth top surface of driving line gear, 3secondary tooth flank of driving line gear, 4driving wheel body, 5tooth flank of driven line gear, 6tooth bottom surface of driven line gear, 7secondary tooth flank of driven line gear, 8driven wheel body, 9driving contact curve, 10first driving tooth thickness auxiliary curve, 11second driving tooth thickness auxiliary curve, 12first driving tooth profile arc, 13driving tooth profile straight line segment, 14second driving tooth profile arc, 15driven contact curve, 16first driven tooth thickness auxiliary curve, 17second driven tooth thickness auxiliary curve, 18first driven tooth profile arc, 19driven tooth profile straight line segment, 20second driven tooth profile arc.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0037] The present invention is further described below in combination with the accompanying drawings, but the implementations of the present invention are not limited hereto.

[0038] Referring to FIG. 1, the present invention provides a concave-convex arc line gear mechanism for parallel shaft transmission, which comprises a driving line gear and a driven line gear, axes of the driving line gear and the driven line gear being parallel to each other to form a transmission pair. The driving line gear is consisted of convex gear teeth and a driving wheel body, a surface of the convex gear tooth comprising a pair of convex arc-shaped tooth flanks and a tooth top surface. The driven line gear is consisted of concave gear teeth and a driven wheel body, a surface of the concave gear tooth comprising a pair of concave arc-shaped tooth flanks and a tooth bottom surface.

[0039] Specifically, the convex tooth of the driving line gear is formed by driving tooth profile composed of two sections of arcs and a section of straight line 13 moving along a driving contact curve 9 and two driving tooth thickness auxiliary curves. The concave tooth of the driven line gear is formed by driven tooth profile composed of two sections of arcs and a section of straight line 19 moving along a driven contact curve 15 and two driven tooth thickness auxiliary curves. The driving contact curve 9 and the driven contact curve 15 are a pair of conjugate space curves which conform to space curve meshing equations. A tooth profile of the convex teeth of the driving wheel and a tooth profile of the concave teeth of the driven line gear are located on normal planes of the driving contact curve 9 and the driven contact curve 15, respectively.

[0040] Specifically, a meshing track of the transmission pair during transmission is a space curve. One arc-shaped tooth flank of the driving line gear and one arc-shaped tooth flank of the driven line gear present a point contact of convex arc and concave arc at a meshing point.

[0041] In addition, according to requirements, tooth shapes of the driving line gear and the driven line gear are interchangeable, i.e., the driving line gear has concave gear teeth, while the driven line gear has convex gear teeth.

[0042] Specifically, the driving wheel body and the driven wheel body are cylindrical wheel bodies. Driving teeth project from the cylindrical wheel body. Driven teeth are recessed into the cylindrical wheel body.

[0043] By means of setting different values of transmission ratio, the present embodiment may be applied in speed-increasing and speed-reducing transmissions. The axes of the driving line gear and the driven line gear are parallel to each other. Number of teeth of the driving line gear is 1, while number of teeth of the driven line gear is 10. A speed-reducing transmission with transmission ratio of 10:1 is realized in a meshing process.

[0044] See FIGS. 1, 2, 3, 4, 7 and 8 for structure of the driving line gear. A tooth flank 1, a driving tooth top surface 2 and a driving line gear secondary tooth flank 3 are distributed on the cylindrical driving wheel body 4. The convex tooth of the driving line gear is formed by a first driving tooth profile arc 12 and a second driving tooth profile arc 14 which are symmetrical and the driving tooth profile straight line segment 13 moving along the driving contact curve 9, a first driving tooth thickness auxiliary curve 10 and a second driving tooth thickness auxiliary curve 11. Arc radius of the first driving tooth profile arc 12 and the second driving tooth profile arc 14 is .sub.1. An angle between a line which connects the meshing point and an arc center and a binormal vector of the driving contact curve 9 is . The driving tooth profile straight line segment 13 and the binormal vector of the driving contact curve 9 are parallel to each other. A distance between the driving tooth profile straight line segment and the binormal vector is h.sub.a1. A-A section plane is the normal plane of the driving contact curve 9 at any point.

[0045] See FIGS. 1, 2, 5, 6, 7 and 8 for structure of the driven wheel. A plurality of gear teeth are distributed on the cylindrical driven wheel body 8. Each of the gear teeth is consisted of a driven tooth flank 5, a driven tooth bottom surface 6 and a driven secondary tooth flank 7. The concave tooth of the driven line gear is formed by a first driven tooth profile arc 18 and a second driven tooth profile arc 20 which are symmetrical and the driven tooth profile straight line segment 19 moving along the driven contact curve 15, a first driven tooth thickness auxiliary curve 16 and a second driven tooth thickness auxiliary curve 17. Arc radius of the first driven tooth profile arc 18 and the second driven tooth profile arc 20 is .sub.2. An angle between a line which connects the meshing point and an arc center and a binormal vector of the driven contact curve 15 is . The driven tooth profile straight line segment 19 and the binormal vector of the driven contact curve 15 are parallel to each other. A distance between the driven tooth profile straight line segment 19 and the binormal vector is h.sub.f2. B-B section plane is the normal plane of the driven contact curve at any point.

[0046] The driving tooth flank 1 is in a point-contact meshing with the driven tooth flank 5 in a transmission process. Referring to FIGS. 2 and 8, on a normal plane P at the meshing point of a pair of driving contact curve 9 and driven contact curve 15, the driving tooth flank 1 is tangent to the driven tooth flank 15. M is a contact point, M.sub.11 is an intersection point of the first driving tooth thickness auxiliary curve 10 of the driving line gear with the normal plane P, M.sub.12 is an intersection point of the second driving tooth thickness auxiliary curve 11 of the driving line gear with the normal plane P, M.sub.21 is an intersection point of the first driven tooth thickness auxiliary curve 16 of the driven line gear with the normal plane P, and M.sub.22 is an intersection point of the second driven tooth thickness auxiliary curve 17 of the driven line gear with the normal plane P. M M.sub.11 is equal to M.sub.11 M.sub.12, with a value being half of tooth thickness of the driving line gear, i.e. c.sub.1. M M.sub.21 is equal to M.sub.21 M.sub.22, with a value being half of tooth thickness of the driven line gear, i.e. c.sub.2. The driving tooth flank 1 and the driving secondary tooth flank 3 are symmetrical about a midperpendicular at a midpoint M.sub.11 of M and M.sub.12, and the driven tooth flank 5 and the driven secondary tooth flank 7 are symmetrical about a midperpendicular at a midpoint M.sub.21 of M and M.sub.22. The gear teeth of the driving line gear and the gear teeth of the driven line gear have the same tooth profile on the normal planes P of respective contact curves.

[0047] Given parameters are set as follows: m=10 mm, n=8 mm, i.sub.12=10, number of teeth of the driving line gear is 1, number of teeth of the driven line gear is 10, the tooth of the driving line gear has a tooth width 2c.sub.1=10 mm, the tooth of the driven line gear has a tooth width 2c.sub.2=14 mm, =35, contact ratio

[00010]$=\frac{.Math..Math.t1}{2.Math.}=1.25,{}_1=6.Math..Math.\mathrm{mm},{}_2=24.Math..Math.\mathrm{mm};$

it can be obtained that a center distance

[00011]$a=\left(1+i_{12}\right).Math.m=110.Math..Math.\mathrm{mm},t\left[-\frac{9.Math.}{4},\frac{}{4}\right],h_{a.Math..Math.1}=2.5.Math..Math.\mathrm{mm},h_{f.Math..Math.2}=3.5.Math..Math.\mathrm{mm},$

the driving wheel body 4 has a diameter d.sub.f1=2m2(h.sub.a1+d.sub.f*)=14 mm, and the driven wheel body 8 has a diameter d.sub.a2=2(am)+2h.sub.a1=205 mm;

[0048] an equation of the driving contact curve 9 in a coordinate system o.sub.1x.sub.1y.sub.1z.sub.1 can be obtained as follows:

[00012]$\begin{array}{cc}\{\begin{array}{c}{x}_M^{\left(1\right)}=10.Math..Math.\cos .Math..Math.t\\ y_M^{\left(1\right)}=10.Math..Math.\sin .Math..Math.t\\ z_M^{\left(1\right)}=8.Math.+8.Math.t\end{array};& \end{array}$

[0049] an equation of the driven contact curve 15 in a coordinate system o.sub.2x.sub.2y.sub.2z.sub.2 can be obtained as follows:

[00013]$\{\begin{array}{c}{x}_M^{\left(2\right)}=100.Math.\cos .Math..Math.\frac{t+}{i_{12}}\\ y_M^{\left(2\right)}=-100.Math..Math.\sin .Math..Math.\frac{t+}{i_{12}}\\ z_M^{\left(2\right)}=8.Math.+8.Math.t\end{array};$

[0050] equations of the first driving tooth thickness auxiliary curve 10 and the second driving tooth thickness auxiliary curve 11 of the driving line gear in the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1 are obtained as follows, respectively:

[00014]$\begin{array}{cc}\{\begin{array}{c}{x}_{M_{11}}^{\left(2\right)}=10.Math.\cos .Math..Math.t.Math.-\frac{40.Math..Math.\sin .Math..Math.t}{\sqrt{164}}\\ y_{M_{11}}^{\left(2\right)}=10.Math..Math.\sin .Math..Math.t+\frac{40.Math..Math.\cos .Math..Math.t}{\sqrt{164}}\\ z_{M_{11}}^{\left(2\right)}=8.Math.+8.Math.t-\frac{50}{\sqrt{164}}\end{array},& \\ \{\begin{array}{c}{x}_{M_{12}}^{\left(2\right)}=10.Math.\cos .Math..Math.t.Math.-\frac{80.Math..Math.\sin .Math..Math.t}{\sqrt{164}}\\ y_{M_{12}}^{\left(2\right)}=10.Math..Math.\sin .Math..Math.t+\frac{80.Math..Math.\cos .Math..Math.t}{\sqrt{164}}\\ z_{M_{12}}^{\left(2\right)}=8.Math.+8.Math.t-\frac{100}{\sqrt{164}}\end{array};& \end{array}$

[0051] equations of the first driven tooth thickness auxiliary curve 16 and the second driven tooth thickness auxiliary curve 17 of the driven line gear in the coordinate system o.sub.2x.sub.2y.sub.2z.sub.2 are obtained as follows, respectively:

[00015]$\begin{array}{cc}\{\begin{array}{c}{x}_{M_{21}}^{\left(2\right)}=100.Math.\cos .Math..Math.\frac{t+}{i_{12}}+\frac{56}{\sqrt{164}}.Math.\sin .Math..Math.\frac{t+}{i_{12}}\\ y_{M_{21}}^{\left(2\right)}=-100.Math.\sin .Math..Math.\frac{t+}{i_{12}}+\frac{56}{\sqrt{164}}.Math.\cos .Math..Math.\frac{t+}{i_{12}}\\ z_{M_{21}}^{\left(2\right)}=8.Math.+8.Math.t+\frac{70}{\sqrt{164}}\end{array},& \\ \{\begin{array}{c}{x}_{M_{22}}^{\left(2\right)}=100.Math.\cos .Math..Math.\frac{t+}{i_{12}}+\frac{112}{\sqrt{164}}.Math.\sin .Math..Math.\frac{t+}{i_{12}}\\ y_{M_{22}}^{\left(2\right)}=-100.Math.\sin .Math..Math.\frac{t+}{i_{12}}+\frac{112}{\sqrt{164}}.Math.\cos .Math..Math.\frac{t+}{i_{12}}\\ z_{M_{22}}^{\left(2\right)}=8.Math.+8.Math.t+\frac{140}{\sqrt{164}}\end{array}.& \end{array}$

[0052] The line gear mechanism of the present invention has high contact strength, bending strength and great rigidity, possesses greater load-bearing capacity, can be machined by numerical control method and is easy to be mass-produced. Minimum number of teeth of the driving line gear is 1, the transmission ratio is higher than that of the existing transmission mechanisms such as spur gear and helical gear, and a single-stage high contact ratio transmission with high transmission ratio may be realized. Compared with spur gear and helical gear of traditional mechanical transmission, a compact construction may greatly save installation space, and thus it is suitable for conventional mechanical applications.

[0053] As described above, the present invention can be well implemented.

[0054] The above-described embodiments of the present invention are just examples for describing the present invention clearly, but not limitation to the implementations of the present invention. For those having ordinary skill in the art, variations or changes in different forms can be made on the basis of the above description. All of the implementations should not and could not be exhaustive herein. Any amendment, equivalent replacement and improvement made within the spirit and principle of the present invention shall all be included within the scope of protection of the claims of the present invention.